Force = (mass) x (acceleration)
Mass = (force) / (acceleration)
The mass of the object is . . .
(force in the direction of acceleration) / (acceleration in that direction) .
If the normal force isn't in the direction of motion, then it isn't involved in the acceleration, and it can be ignored.
(In more-complicated problems, the normal force might produce a friction force
along the direction of motion, and that would get involved with the applied force.)